Question

$$sqrt4{1250x^2}$$

Explanation:

Step1: Factorize 1250

First, we factorize 1250. We know that \(1250 = 2\times625=2\times5^4\). So the expression becomes \(\sqrt[4]{2\times5^4\times x^2}\).

Step2: Apply the fourth - root property

The fourth - root property is \(\sqrt[n]{ab}=\sqrt[n]{a}\times\sqrt[n]{b}\) (for \(a\geq0,b\geq0\) when \(n\) is even) and \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\) (when \(a\geq0\)).
Using \(\sqrt[4]{ab}=\sqrt[4]{a}\times\sqrt[4]{b}\), we can split \(\sqrt[4]{2\times5^4\times x^2}\) into \(\sqrt[4]{5^4}\times\sqrt[4]{2x^2}\).
Since \(\sqrt[4]{5^4} = 5\) (because \((5)^4=5^4\) and the fourth - root of \(a^4\) is \(a\) for \(a\geq0\)), the expression simplifies to \(5\sqrt[4]{2x^2}\).
We can also rewrite \(x^2\) as \(x^{4\times\frac{1}{2}}\), so \(\sqrt[4]{x^2}=x^{\frac{2}{4}} = x^{\frac{1}{2}}=\sqrt{x}\) (for \(x\geq0\)). So another form of the simplified expression is \(5\sqrt[4]{2}\sqrt{x}\) or \(5x^{\frac{1}{2}}\sqrt[4]{2}\).

Answer:

\(5\sqrt[4]{2x^{2}}\) (or equivalent forms like \(5\sqrt{x}\sqrt[4]{2}\))